Introduction

Abstract

The chapter contains an overview of the main new results in the book: a spectral characterisation of covering equivalence of Riemannian manifolds using spectra of finitely many twisted Laplacians, a more detailed version for non-arithmetic Riemann surfaces, a version using the length spectrum. It also contains a spectral characterisation of weak conjugacy. The chapter contains pointers forward to other chapters that contain an exposition of the basic tools and methods, as well as detailed constructions, proofs and examples.

The chapter contains an overview of the main new results in the book: a spectral characterisation of covering equivalence of Riemannian manifolds using spectra of finitely many twisted Laplacians, a more detailed version for non-arithmetic Riemann surfaces, a version using the length spectrum. It also contains a spectral characterisation of weak conjugacy. The chapter contains pointers forward to other chapters that contain an exposition of the basic tools and methods, as well as detailed constructions, proofs and examples.

1.1 Setup and Conditions

Let M ₁ and M ₂ be a pair of connected closed oriented smooth Riemannian manifolds. There exist such M ₁ and M ₂ that are not isometric, but isospectral, i.e., they have the same Laplace spectrum with multiplicities. This leaves open the question whether equality of spectra of other geometrically defined operators on M ₁ and M ₂ is equivalent to M ₁ and M ₂ being isometric. In this text we investigate the use of twisted Laplacians (acting on sections of flat bundles constructed from representations of fundamental groups) in answering this question. We will see that under two conditions on M ₁ and M ₂, equality of finitely many suitably twisted Laplace spectra implies isometry of the manifolds. That at least some condition is necessary for such a result to hold is illustrated by the existence of simply connected isospectral non-isometric manifolds [87].

The first condition is the following: we suppose that the manifolds M ₁ and M ₂ are finite Riemannian coverings of a developable Riemannian orbifold M ₀ (meaning that the universal covering of M ₀ is a manifold), expressed through a diagram

(1.1)

Such a diagram may be extended to a diagram of finite coverings

(1.2)

where M is a connected closed smooth Riemannian manifold M with $q_1 \colon M \twoheadrightarrow M_1:=H_1 \backslash M$, $q_2 \colon M \twoheadrightarrow M_2:=H_2 \backslash M$ and $q \colon M \twoheadrightarrow M_0:=G\backslash M$ Galois covers (Proposition 2.4.1); in particular, M ₁ and M ₂ are commensurable. Commensurability is in general weaker than the existence of a diagram (1.1) (see Proposition 2.5.4). However, if M ₁ and M ₂ are commensurable hyperbolic manifolds, then a diagram (1.1) (with an orbifold M ₀) exists if the corresponding lattices are not arithmetic (Proposition 2.5.2).

The second condition is related to the action of G on the first homology group of M. Let F _ℓ denote the field with ℓ elements. In terms of the data in diagram (1.2), we require that for some prime number ℓ not dividing |G|, the F _ℓ[G]-module $\operatorname {H}_1(M,{\mathbf {F}}_\ell ) = \operatorname {H}_1(M) \otimes _{\mathbf {Z}} {\mathbf {F}}_\ell$ contains the permutation representation of G acting by left multiplication on the cosets G∕H _i for either i = 1 or i = 2 (or both), i.e., $(\operatorname {Ind}_{H_i}^G \mathbf {1}) \otimes _{\mathbf {Z}} {\mathbf {F}}_\ell$ is an F _ℓ[G]-submodule of $\operatorname {H}_1(M,{\mathbf {F}}_\ell )$. Concretely, this means that if there are n cosets, then the F _ℓ-vector space $\operatorname {H}_1(M,{\mathbf {F}}_\ell )$ contains n linear independent vectors that are permuted in the same way as those n cosets under the action of any g ∈ G.

This second condition is implied by the stronger requirement that the G-action is F _ℓ-homologically wide, where, for a general field K, we say that the G-action is K-homologically wide if the regular representation K[G] of G occurs in the homology representation $G \rightarrow \operatorname {Aut}(\operatorname {H}_1(M,K))$ (Lemma 8.1.2). This condition has the advantage of no longer referring to M ₁ and M ₂ (or H ₁ and H ₂).

An even stronger, geometrically tangible, condition is that the action of G on M is Q-homologically wide; this means that there is a free homology class ω on M such that the elements {gω: g ∈ G} are linearly independent in $\operatorname {H}_1(M,\mathbf {Q})$. This is diametric to the condition of homologically trivial group actions found more frequently in the literature (see, e.g., [99]). A Q-homologically wide action is F _ℓ-homologically wide for any ℓ coprime to |G| (Lemma 8.2.1).

In Chap. 9, we discuss Q-homological wideness for certain low dimensional manifolds and locally symmetric spaces. For non-trivial fixed-point free group actions on orientable surfaces, Q-homologically wideness is equivalent to any of the spaces M, M ₀, M ₁ or M ₂ having negative Euler characteristic (Proposition 9.1.1). A (not necessarily fixed point free) holomorphic group action on a Riemann surface M is Q-homologically wide if the Euler characteristic of the quotient surface M ₀ satisfies $\chi _{M_0}<0$ (Proposition 9.1.2). In higher dimension, the picture can vary widely: for any n ≥ 3, we use standard surgery methods to construct an n-manifold with a free action by any given non-trivial group that is or is not Q-homologically wide (Proposition 9.3.1 and Corollary 9.3.3). Since locally symmetric spaces of rank ≥ 2 have trivial rational homology, no non-trivial group action on them can be Q-homologically wide (see Sect. 9.4). On the other hand, in rank one, the condition relates to decomposition results for automorphic representations (see Sect. 9.5). A disadvantage of Q-homological wideness is that it discards torsion homology; in Sect. 9.7 we give an example of some non-trivial F ₅-homologically wide group actions on the Seifert–Weber dodecahedral space (that has homology $\operatorname {H}_1(M,\mathbf {Z}) \cong \left ({\mathbf {Z}}/{5}{\mathbf {Z}}\right )^3$).

1.2 Overview of Main Results

Our results involve spectra of twisted Laplace operators Δ_ρ corresponding to unitary representations $\rho \colon \pi _1(M) \rightarrow \operatorname {U}(N, \mathbf {C})$; these are symmetric second order elliptic differential operators acting on sections of flat bundles E _ρ over M; the sections are conveniently described as smooth vector valued functions on the universal cover that are equivariant with respect to the representation, and on these, Δ_ρ acts (componentwise) like the usual Laplacian of the universal cover (cf. Chap. 3). In fact, our representations will factor through a specific finite group, allowing for a very concrete description of the operators (cf. Sects. 3.6 and 3.7). Denote the spectrum of such an operator by σ _M(ρ), where the index M indicates that the Laplacian is defined on sections over the space M.

Returning to the setup in diagram (1.1), we call M ₁ and M ₂ equivalent Riemannian covers of M ₀ if there is an isometry between them induced by a conjugacy of the fundamental groups of M ₁ and M ₂ inside that of M ₀ (since M ₀ is developable, its universal covering is a manifold and we mean the subgroup of its isometry group that fixes M ₀ pointwise, see Lemma 2.1.2). Our main result states that in this situation isometry can be detected via the spectra of finitely many specific twisted Laplacians.

Theorem 1.2.1

Suppose we have a diagram (1.1)and suppose that the action of G on M in the extended diagram (1.2)is F _ℓ -homologically wide for some prime ℓ coprime to |G|. Then M ₁ and M ₂ are equivalent Riemannian covers of M ₀ ( in particular, isometric)if and only if the multiplicity of zero in the spectra of a finite number of specific twisted Laplacians on M ₁ and M ₂ is equal. In fact, at most 2ℓ|Hom(H ₂, C ^∗)| equalities suffice.

For a precise formulation of the required twists and a more technical condition (denoted (∗)) that is weaker than homological wideness, the reader is encouraged to glance at Theorem 6.4.1, which is a more detailed formulation of Theorem 1.2.1, our main result. The detailed formulation reveals that the representations occurring in the theorem are constructed explicitly using induction and restriction of special characters (termed “solitary” below) on groups corresponding to a specific finite Riemannian covering of M, whose existence is guaranteed by the assumption of homological wideness (or the weaker requirement (∗)). Riemannian equivalence over M ₀ is the same as conjugacy of H ₁ and H ₂ in G; a merit of the theorem is to show that this can be verified via a spectral geometric criterion using twists, where the twists on M ₁ are constructed using information from M ₂ and vice versa.

Sunada [90] showed that if we are given a diagram of the form (1.2), and H ₁ and H ₂ are weakly conjugate (meaning that the permutation representations given by the action of G by left multiplication on the cosets G∕H ₁ and G∕H ₂ are isomorphic), M ₁ and M ₂ are isospectral for the Laplace operator. The following example illustrates our theorem in such a situation.

Example 1.2.2

We provide the explicit data for what is maybe the oldest example of weakly conjugate subgroups, due to Gaßmann [38]:

$$\displaystyle \begin{aligned} G=S_6,\ H_1 = \langle (12)(34), (13)(24) \rangle,\ H_2=\langle (12)(34), (12)(56) \rangle, \end{aligned}$$

with both H ₁ and H ₂ isomorphic to the Klein four-group, but not conjugate inside S ₆ [38, pp. 674–675]. As in [90, p. 174], choose a compact Riemann surface M ₀ of genus 2 and a surjective group homomorphism $\pi _1(M_0) \twoheadrightarrow G$. This leads to a diagram of the form (1.2) [90, §2], and homological wideness is immediate from Proposition 9.1.1. In this case, M ₁ and M ₂ are inequivalent covers of M ₀, but they have the same Laplace spectrum [90, §2]. We can set ℓ = 7 in the main theorem, and, with the group G having order 720, inequivalence of M ₁ and M ₂ may be verified purely spectrally by checking 56 equalities of multiplicities of zero in the spectrum of twisted Laplacians corresponding to representations of dimension 180. $\hfill \lozenge$

To show the scope of our result, we discuss several more examples in Chap. 11. Our result should be contrasted with [4, Thm. 1.1], where it is shown that large non-arithmetic hyperbolic manifolds M admit arbitrary large sets of (strongly) isospectral but pairwise non-isometric finite Riemann coverings.

Our method of proof for Theorem 1.2.1, presented in Chaps. 2–6, is based on a similar construction of Solomatin [89] for algebraic function fields, which in turn is based on number theoretical work of Bart de Smit in [28]. The analogy between number theory and spectral differential geometry was pioneered by Sunada [90] (see also the survey [93]), and the importance of representation theoretical techniques was pointed out early on by Sunada [91] and Pesce [79]. We have given a self-contained presentation, with references to the number theory literature when appropriate. Our construction uses a certain wreath product of G with a cyclic group; in Sect. 7.4, we describe a universality property of such wreath products that should make their appearance look less surprising.

We have formulated our results using spectra, but the analogy to number theory becomes most apparent by instead using spectral zeta functions of (twisted) Laplacians; that this is equivalent is explained in Sects. 3.2 and 3.10, pointing to some subtleties concerning the multiplicity of zero in the spectrum of more general operators.

From our earlier brief discussion of the two conditions in the theorem, we find the following specific result in dimension two.

Corollary 1.2.3

Let M ₁, M ₂ be two commensurable non-arithmetic closed Riemann surfaces. Then they admit a diagram (1.1) and, assuming the corresponding orbifold M ₀ satisfies $\chi _{M_0} < 0$ , isometry of M ₁ and M ₂ can be checked by computing the multiplicity of zero in at most $4 ((\chi _{M_1}\chi _{M_2}/(\chi ^{\mathrm {orb}}_{M_0})^2)!)^2$ twisted Laplace spectra, where $\chi ^{\mathrm {orb}}_{M_0}$ is the orbifold Euler characteristic, defined in (9.1).

The detailed formulation and proof can be found in Corollary 9.1.4. If one believes in a positive answer to the (open since several decades) question whether commensurability of such Riemann surfaces is implied by their isospectrality, then a purely spectral formulation of the corollary is possible, replacing “commensurable” by “isospectral”. Intriguingly, the only cases where a positive answer to the open question is known are arithmetic [81], precisely the ones excluded in the corollary.

In Chap. 12, we study the analogue of Theorem 1.2.1 for the length spectrum on negatively curved manifolds. The statement about agreement of multiplicities of zero in certain spectra is changed into equality of the pole order of certain L-series (details are found in Theorem 12.2.4).

Theorem 1.2.4

Suppose we have a diagram (1.1)of negatively curved Riemannian manifolds with (common) volume entropy h, and suppose that the action of G on M in the extended diagram (1.2)is F _ℓ -homologically wide for some ℓ coprime to |G|. Then M ₁ and M ₂ are equivalent Riemannian covers of M ₀ if and only if the pole order at s = h of a finite number of specific L-series of representations on M ₁ and M ₂ is equal.

Sunada’s result [90] quoted above says that weak conjugacy implies isospectrality, but the converse is not necessarily true; this leaves open the question to characterise weak conjugacy of H ₁ and H ₂ in G in a spectral way using the associated manifolds M ₁ and M ₂. One of our intermediate results answers this question using induction and restriction from the trivial representation 1.

Proposition 1.2.5

If we have a diagram (1.2), then H ₁ and H ₂ are weakly conjugate if and only if the multiplicity of the zero eigenvalue in $\sigma _{M_i}(\operatorname {Res}_{H_i}^G \operatorname {Ind}_{H_j}^G \mathbf {1})$ is independent of i, j = 1, 2.

This result, reformulated in Proposition 4.2.4, is proven by an adaptation of a number theoretical argument of Nagata [73]. The crucial differential geometric ingredient is the spectral characterisation of the multiplicity of the trivial representation in any given representation (Lemma 3.9.1). As a corollary, we get a spectral characterisation of strong isospectrality (cf. Definition 4.2.2 and Corollary 4.2.5).

Open Problem

The most pressing question that remains open concerns the case where there is not necessarily a common finite orbifold quotient: are twisted isospectral manifolds M ₁ and M ₂ (meaning that there is a bijection of all unitary representations of their fundamental groups such that the spectra of the corresponding twisted Laplacians are equal) with large fundamental groups (i.e., containing a finite index subgroup with a non-abelian free group as quotient) isometric?

Project

Develop the theory when M ₁ and M ₂ are also orbifolds.

Project

Develop an analogous theory for graphs instead of manifolds.